The field of mathematics education is undergoing radical changes. Mathematics has been traditionally taught in a manner where the teacher tells the students exactly what to do and then the students do it. This leads to a narrow understanding of the field of mathematics as well as its applicability in the real world. The traditional method is easier for the teacher, since it is typified by one-step, rote processes that are easy to teach and easy to grade. This situation is referred to as a "teacher centered classroom". This traditional method is often implemented by the use of worksheets containing repetitive, one-step arithmetic problems.
The result of traditional teaching has been dismal as far as preparing students for real world situations where mathematics is used. The United States has continually seen falling SAT scores, low ranking in mathematics ability among industrialized nations, and complaints among employers that graduates from the nation's schools are mathematically illiterate.
In 1989, the NCTM (National Council of Teachers of Mathematics) issued a set of recommendations referred to as "The Standards", in which they advocated the teaching of mathematics in a way that reflects problem solving in the real world, especially that problem solving which meets the following criteria: hands-on investigation, estimation, reasoning, experimentation, multiple solution steps, variable methods, and group work. Additional criteria require the demonstration of a student's ability to communicate mathematically via charts, graphs, diagrams, discussions, and writings. Current state and federal guidelines for the teaching of mathematics have incorporated most of these NCTM recommendations into their own directives.
Specifically, the NCTM recommends questions that demand reasoning in order to get an answer; methods that encourage group work, trial-and-error, and discussion; and answers that contain an explanation rather than a single number. The NCTM wants to get away from rote memorization of rules and teaching by telling, and move on to investigating questions from problem situations and connecting mathematics to other subjects and to the world outside the classroom.
One of the difficulties of the NCTM recommendations is the required culture shift in the teaching of mathematics. Generations of teachers have grown up in the classroom culture where the teacher tells the student what to do, mathematics is all done in one (or two) rote steps, and there is little connection to the outside world. Teachers growing up in this environment are not comfortable with experimenting, estimating, and reasoning. Quite frankly, mathematics is typically their least favorite subject and is done as quickly as possible. A major flaw in the NCTM recommendations is that no vehicle was supplied to meet these standards. Teachers can buy dice, rulers, geometric shapes, marbles, little sticks, and a multitude of other manipulatives to work with. Many teachers are at a loss as to what to do with these manipulatives, due to a lack of basic mathematics training. On visiting an elementary classroom, one often sees children using these manipulatives to build houses or to play with in some other inappropriate way.